Question

Given is the right triangle  ABC  with altitude  CO . Which one of the following statements is true and contributes to the proof of the Pythagorean Theorem? (1 point) Responses △AOC∼△BOC triangle upper A upper B upper C congruent to triangle upper A upper C upper O △ABC∼△AOC triangle upper A upper B upper C congruent to triangle upper A upper C upper O △ABC∼△BOC triangle upper A upper B upper C congruent to triangle upper A upper C upper O △AOC∼△COB

Answer 1

To determine which statement contributes to the proof of the Pythagorean Theorem, we consider the relationships between the triangles formed in right triangle \( ABC \) with altitude \( CO \).

In right triangle \( ABC \):

1. Triangle \( AOC \) is similar to triangle \( ABC \) because they share angle \( A \) and both have a right angle.
2. Triangle \( BOC \) is also similar to triangle \( ABC \) because they share angle \( B \) and both have a right angle.
3. The similarity of these triangles allows us to set up proportions which lead to establishing relationships that ultimately lead to the Pythagorean Theorem.

Given the choices, the statement that is true and contributes to the proof of the Pythagorean Theorem is:

• \( \triangle ABC \sim \triangle AOC \)

This similarity leads to the relationships:

• \( \frac{AC}{AB} = \frac{AO}{AC} \)
• \( \frac{BC}{AB} = \frac{BO}{BC} \)

Similar triangles are key in deriving the Pythagorean theorem.

So, the statement \( \triangle ABC \sim \triangle AOC \) is the one that is true and is used in the proof of the Pythagorean Theorem.